User:Egm6321.f10.team4.Yoon/Mtg36

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_09_15_00_14.djvu | Mtg 36:]] Tue, 9 Nov 10

[[media: 2010_11_09_15_00_14.djvu | Page 36-1]]
Note:

Axisymmetry problem [[media: 2010_11_04_14_51_31.djvu | p.34-2]].

Q: what if axis of symmetry is arbitrarily oriented (i.e. no longer the South-to-North axis)?

A: Change the coordinate system such the new S-to-N axis coincides with new axis of symmetry => Same equation

End Note

= Heat conduction on a sphere (cont'd) =

[[media: 2010_11_05_07_46_02.djvu | Eqn(0) p.35-2, Eqn(3) p.35-3 ]]

[[media: Egm6321.f09.mtg31.djvu | F09 p.31-2]]: solve quadratic equation (1) to obtain (2) and (3).

[[media: 2010_11_09_15_00_14.djvu | Page 36-2]]
[[media: 2010_11_05_07_46_02.djvu | Eqn(1) p.35-1 ]]

Superposition: Now 2nd seperated equation [[media: 2010_11_05_07_46_02.djvu | Eqn(3) p.34-4 ]](Legendre equation if k=n(n+1)) Family of 1st homogeneous Solutions = Legndre Polynomials

Some problems to solve...

[[media: 2010_11_09_15_00_14.djvu | Page 36-3]]
Back to heat equation on a sphere...

Family of 2nd homogeneous solutions $$\displaystyle \left\{ Q_0, Q_1, Q_2, ...\right\}$$(see [[media: 2010_11_02_14_58_42.djvu | p.32-2 ]])

For each n: with $$\displaystyle \mu := \sin \theta $$ from [[media: 2010_11_05_07_46_02.djvu | Eqn(1) p.35-2 ]].

where $$\displaystyle R_n(r)$$ is given in (1) p.36-2, and $$\displaystyle \Theta_n(\theta)$$ is given in (1) above.